Scheduling concurrent production over a finite planning horizon: polynomially solvable cases

The paper analyzes a manufacturing system made up of one workstation which is able to produce concurrently a number of product types with controllable production rates in response to time-dependent product demands. Given a finite planning horizon, the objective is to minimize production cost, which is incurred when the workstation is not idle and inventory and backlog costs, which are incurred when the meeting of demand results in inventory surpluses and shortages. With the aid of the maximum principle, optimal production regimes are derived and continuous-time scheduling is reduced to a combinatorial problem of sequencing and timing the regimes. The problem is proved to be polynomially solvable if demand does not exceed the capacity of the workstation or it is steadily pressing and the costs are “agreeable”.